It is currently 24 Jun 2017, 05:36

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte

Author Message
TAGS:

Hide Tags

Senior Manager
Joined: 01 Feb 2005
Posts: 271
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

15 Dec 2009, 15:48
17
KUDOS
58
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

23% (05:03) correct 77% (02:30) wrong based on 1504 sessions

HideShow timer Statistics

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: nonnegative integers - MGMAT Challenge [#permalink]

Show Tags

15 Dec 2009, 17:04
23
KUDOS
Expert's post
22
This post was
BOOKMARKED
axl_oz wrote:
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

Suppose you got the answer of 2 for the values of $$x$$ and $$y$$ as 4 and 2.

$$2^4+2^2=4^2+2^2$$ --> $$|4-2|=2$$

But if we check for $$y=0$$, we'll get:

$$2^x+2^0=x^2+0^2$$ --> $$2^x+1=x^2$$ --> $$2^x=(x-1)(x+1)$$ --> $$x=3$$

$$2^3+2^0=9=3^2+0^2$$

$$|x-y|=|3-0|=3$$

4 can not be the greatest value as when you increase $$x$$ so as $$x-y$$ to be $$4$$, $$2^x+2^y$$ will always be more than $$x^2+y^2$$.
_________________
Intern
Joined: 18 Dec 2009
Posts: 13
Re: nonnegative integers - MGMAT Challenge [#permalink]

Show Tags

18 Dec 2009, 20:01
good one! i tried backsolving and started with the middle one and then checked one above and one below. got the result.
Manager
Joined: 09 May 2009
Posts: 204
Re: nonnegative integers - MGMAT Challenge [#permalink]

Show Tags

18 Dec 2009, 20:51
it should be 3 , but my way is hit and trial
_________________

GMAT is not a game for losers , and the moment u decide to appear for it u are no more a loser........ITS A BRAIN GAME

Intern
Joined: 12 Oct 2009
Posts: 16
Re: nonnegative integers - MGMAT Challenge [#permalink]

Show Tags

19 Dec 2009, 07:20
Great Problem! Since your trying to find the greatest value of X-Y, you just have to assume that Y=0, like Bunel said and then use the "hit and trial" approach like xcusem... Said. The algebratic approach is great too, but I know for me personally it opens up the opportunity for me to make silly mistakes. So I try to not use it unless necessary.

Posted from my mobile device
Intern
Joined: 15 Jan 2013
Posts: 39
Concentration: Finance, Operations
GPA: 4
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

08 Feb 2013, 09:27
7
KUDOS
3
This post was
BOOKMARKED
axl_oz wrote:
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

Since we need to maximize the value of |x – y|, we can do that in two ways...1)make y negative, which is not possible as per the question...2)make y= 0..putting y=0 you will get an equation in x and on hit and trial method u will get the value of x as 3, which will satisfy the equation....
putting x=3 and y=0, we will get the value of |x – y| as 3.
Intern
Joined: 22 Sep 2012
Posts: 8
Concentration: Entrepreneurship, Strategy
GPA: 3.3
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

20 Mar 2013, 12:16
I have a question for Bunuel: we know that |x−y| = ((x-y)*2)*1/2 = (x*2+y*2-2xy)*1/2

so substituting the value of x*2 +y*2 as 2*x + 2*y in the above equation

I got |x−y| = (2*x + 2*y -2xy)*1/2

then, substituting values for x (taking y=0)...the max value for x can be anything more than 0, coz if you take (x=4) then u'l end up with |x−y| = 4

please can you help me with this problem, where did I go wrong???
Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

21 Mar 2013, 03:39
1
KUDOS
Expert's post
Ace99 wrote:
I have a question for Bunuel: we know that |x−y| = ((x-y)*2)*1/2 = (x*2+y*2-2xy)*1/2

so substituting the value of x*2 +y*2 as 2*x + 2*y in the above equation

I got |x−y| = (2*x + 2*y -2xy)*1/2

then, substituting values for x (taking y=0)...the max value for x can be anything more than 0, coz if you take (x=4) then u'l end up with |x−y| = 4

please can you help me with this problem, where did I go wrong???

I don't understand the red part above at all...
_________________
Intern
Joined: 22 Sep 2012
Posts: 8
Concentration: Entrepreneurship, Strategy
GPA: 3.3
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

21 Mar 2013, 07:35
sorry...what i meant to say was, if you took the eq: |x−y| = (2*x + 2*y -2xy)*1/2

and substitute the values (x = 4 & y = 0), then we'll end up with |x−y| = (17)*1/2 (which is almost equal to 4)

but before u mentioned the max value of |x−y| = 3.

did i do something wrong??
Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

21 Mar 2013, 08:19
Ace99 wrote:
sorry...what i meant to say was, if you took the eq: |x−y| = (2*x + 2*y -2xy)*1/2

and substitute the values (x = 4 & y = 0), then we'll end up with |x−y| = (17)*1/2 (which is almost equal to 4)

but before u mentioned the max value of |x−y| = 3.

did i do something wrong??

First of all, I think you mean 2^x + 2^y -2xy rather than 2*x + 2*y -2xy.

Next, x=4 and y=0 does NOT satisfy 2^x + 2^y = x^2 + y^2, thus these values are not possible.

Hope it's clear.
_________________
Manager
Joined: 14 Aug 2005
Posts: 84
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

21 Mar 2013, 09:34
Brunel and all,

Is it a rule to apply one value as zero whenever it is given:

1) Both x and y are non-negative integers
2) we need to find the max value of x-y

What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise?
_________________

One Last Shot

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

21 Mar 2013, 21:56
12
KUDOS
Expert's post
5
This post was
BOOKMARKED
surya167 wrote:
Brunel and all,

Is it a rule to apply one value as zero whenever it is given:

1) Both x and y are non-negative integers
2) we need to find the max value of x-y

What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise?

Usually, when you are checking for numbers, you do check for 0. It's often a transition point for patterns. Secondly, the question used the term 'non-negative integers' instead of 'positive integers' - this means 0 would probably have a role to play.
There are no such rules but common sense says that we must not ignore 0.

Also, this question doesn't really test max min concepts. It is a direct application of your understanding of exponential relations discussed in the post: http://www.veritasprep.com/blog/2013/01 ... cognition/

Now, when we look at the equation, 2^x + 2^y = x^2 + y^2, some things come to mind:
1. It is not very easy to find values that satisfy this equation.
2. But there must be some values which satisfy since we are looking for a value of |x – y|
3. If x = y = 2, the equation is satisfied since all terms become equal and |x – y| = 0 which is the minimum value of |x – y|.

Usually, the left hand side will be greater than the right hand side (as discussed in the post, 2^n will usually be greater than x^2 except in very few cases). So we must focus on those 'very few cases'. Also, we need to make x and y unequal.

We know (from the post) that 2^4 = 4^2 is one solution so we could put x = 4 while keeping y = 2. The equation will be satisfied and |x – y| = 2

Now, we also know that 2^x < x^2 when x = 3. So is there a solution there as well? The difference between 2^3 and 3^2 is of 1 so can we create a difference of 1 between the other two terms? Sure! If y = 0, then 2^0 = 1 but 0^2 = 0.
So another solution is 2^3 + 2^0 = 3^2 + 0^2.
Here, |x – y| = 3 which is the maximum difference.

The reason we can be sure that there are no other values is that as you go ahead of 4 on the number line, 2^n will be greater than n^2 (again, discussed in the post). So both left hand side terms will be greater than the right hand side terms i.e. 2^x > x^2 and 2^y > y^2. So, for no other values can we satisfy this equation.

_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Manager
Joined: 16 Jan 2011
Posts: 103
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

14 Oct 2013, 12:10
2
KUDOS
the expression |x-y| to reach its maximum we need y to be 0. Hence, we need to find X.

therefore, 2^x+2^y=x^2+^2 --> 2^x+1=x^2 what means that x is odd. Only 3 satisfies this equation: 2^3+1=3^2.
Hence, x must be equal 3
Intern
Joined: 25 Oct 2013
Posts: 21
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

06 Dec 2013, 00:37
axl_oz wrote:
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

|x – y| for x and y 2^x + 2^y = x^2 + y^2
A. x= y x=y=1 2 + 2 = 1 + 1 wrong
B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong
C. 2 x=3, y=1 8 + 2 = 9 + 1 right
D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong
E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

06 Dec 2013, 02:52
misanguyen2010 wrote:
axl_oz wrote:
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

|x – y| for x and y 2^x + 2^y = x^2 + y^2
A. x= y x=y=1 2 + 2 = 1 + 1 wrong
B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong
C. 2 x=3, y=1 8 + 2 = 9 + 1 right
D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong
E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

_________________
Intern
Joined: 25 Oct 2013
Posts: 21
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

06 Dec 2013, 10:47
1
This post was
BOOKMARKED
Bunuel wrote:
misanguyen2010 wrote:
axl_oz wrote:
If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

|x – y| for x and y 2^x + 2^y = x^2 + y^2
A. x= y x=y=1 2 + 2 = 1 + 1 wrong
B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong
C. 2 x=3, y=1 8 + 2 = 9 + 1 right
D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong
E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

I explained what i confused. Of course I read previous answers and all chose D.
However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer.
Math Expert
Joined: 02 Sep 2009
Posts: 39626
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

07 Dec 2013, 05:52
1
KUDOS
Expert's post
misanguyen2010 wrote:
Bunuel wrote:
misanguyen2010 wrote:
|x – y| for x and y 2^x + 2^y = x^2 + y^2
A. x= y x=y=1 2 + 2 = 1 + 1 wrong
B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong
C. 2 x=3, y=1 8 + 2 = 9 + 1 right
D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong
E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

I explained what i confused. Of course I read previous answers and all chose D.
However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer.

To get the greatest value of |x-y| as 3 consider x=3 and y=0. Notice that these values satisfy $$2^x + 2^y = x^2 + y^2$$ --> $$2^3 + 2^0 =9= 3^2 + 0^2$$.

Hope it helps.
_________________
Intern
Joined: 11 Dec 2013
Posts: 19
Location: India
Concentration: Finance, Technology
Schools: ISB '18
GMAT Date: 03-15-2015
WE: Operations (Telecommunications)
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

12 Dec 2013, 02:02
For x>4,y>4
2^x > x^2 ||| 2^y > y^2
Hence maximum we need to check till (x,y) = {0,1,2,3}

By hit and trial .

For x=3,y=1 and x=4,y=2
|x-y| = 2

Draw the graph for 2^x and x^2. The solution becomes simpler .
Senior Manager
Status: Student
Joined: 26 Aug 2013
Posts: 256
Location: France
Concentration: Finance, General Management
Schools: EMLYON FT'16
GMAT 1: 650 Q47 V32
GPA: 3.44
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

28 Dec 2013, 04:25
Here is how I done it:

1) If |x-y| needs to be max then Y=0, because Y² is only positive
2) Check the answers, those are only integers, you are therefore looking for an integer
3) You have the equation 2^x +1 = X²
4) Use the different choices and you will see that only 3 matches.

Hope it helps!
_________________

Think outside the box

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15939
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]

Show Tags

17 Feb 2015, 07:44
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte   [#permalink] 17 Feb 2015, 07:44

Go to page    1   2    Next  [ 30 posts ]

Similar topics Replies Last post
Similar
Topics:
(2 + x)(2 + y) – (2 + x) – (2 + y) = 3 01 Jun 2017, 17:28
3 If x^2 + y^2 = a and x^2 − y^2 = b, then (2xy)^2= 2 27 Apr 2017, 17:10
39 If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? 13 19 Feb 2017, 02:21
17 If 2x^2 –y^2 = 2xy, then (x+y)^2= 9 31 Aug 2016, 11:27
16 If 2x^2 - y^2 = 2xy, then (x+y)^2 = 8 24 Apr 2017, 20:38
Display posts from previous: Sort by